SRC/sstemr.f(3)

NAME
SRC/sstemr.f

SYNOPSIS

  Functions/Subroutines
subroutine sstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
SSTEMR

Function/Subroutine Documentation

  subroutine sstemr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSTEMR

Purpose:

    !>
    !> SSTEMR computes selected eigenvalues and, optionally, eigenvectors
    !> of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
    !> a well defined set of pairwise different real eigenvalues, the corresponding
    !> real eigenvectors are pairwise orthogonal.
    !>
    !> The spectrum may be computed either completely or partially by specifying
    !> either an interval (VL,VU] or a range of indices IL:IU for the desired
    !> eigenvalues.
    !>
    !> Depending on the number of desired eigenvalues, these are computed either
    !> by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
    !> computed by the use of various suitable L D L^T factorizations near clusters
    !> of close eigenvalues (referred to as RRRs, Relatively Robust
    !> Representations). An informal sketch of the algorithm follows.
    !>
    !> For each unreduced block (submatrix) of T,
    !>    (a) Compute T - sigma I  = L D L^T, so that L and D
    !>        define all the wanted eigenvalues to high relative accuracy.
    !>        This means that small relative changes in the entries of D and L
    !>        cause only small relative changes in the eigenvalues and
    !>        eigenvectors. The standard (unfactored) representation of the
    !>        tridiagonal matrix T does not have this property in general.
    !>    (b) Compute the eigenvalues to suitable accuracy.
    !>        If the eigenvectors are desired, the algorithm attains full
    !>        accuracy of the computed eigenvalues only right before
    !>        the corresponding vectors have to be computed, see steps c) and d).
    !>    (c) For each cluster of close eigenvalues, select a new
    !>        shift close to the cluster, find a new factorization, and refine
    !>        the shifted eigenvalues to suitable accuracy.
    !>    (d) For each eigenvalue with a large enough relative separation compute
    !>        the corresponding eigenvector by forming a rank revealing twisted
    !>        factorization. Go back to (c) for any clusters that remain.
    !>
    !> For more details, see:
    !> - Inderjit S. Dhillon and Beresford N. Parlett:
    !>   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
    !> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol. 25,
    !>   2004.  Also LAPACK Working Note 154.
    !> - Inderjit Dhillon: ,
    !>   Computer Science Division Technical Report No. UCB/CSD-97-971,
    !>   UC Berkeley, May 1997.
    !>
    !> Further Details
    !> 1.SSTEMR works only on machines which follow IEEE-754
    !> floating-point standard in their handling of infinities and NaNs.
    !> This permits the use of efficient inner loops avoiding a check for
    !> zero divisors.
    !>

Parameters
    JOBZ

    !>          JOBZ is CHARACTER*1
    !>          = 'N':  Compute eigenvalues only;
    !>          = 'V':  Compute eigenvalues and eigenvectors.
    !>

    RANGE

    !>          RANGE is CHARACTER*1
    !>          = 'A': all eigenvalues will be found.
    !>          = 'V': all eigenvalues in the half-open interval (VL,VU]
    !>                 will be found.
    !>          = 'I': the IL-th through IU-th eigenvalues will be found.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, the N diagonal elements of the tridiagonal matrix
    !>          T. On exit, D is overwritten.
    !>

    E

    !>          E is REAL array, dimension (N)
    !>          On entry, the (N-1) subdiagonal elements of the tridiagonal
    !>          matrix T in elements 1 to N-1 of E. E(N) need not be set on
    !>          input, but is used internally as workspace.
    !>          On exit, E is overwritten.
    !>

    VL

    !>          VL is REAL
    !>
    !>          If RANGE='V', the lower bound of the interval to
    !>          be searched for eigenvalues. VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    VU

    !>          VU is REAL
    !>
    !>          If RANGE='V', the upper bound of the interval to
    !>          be searched for eigenvalues. VL < VU.
    !>          Not referenced if RANGE = 'A' or 'I'.
    !>

    IL

    !>          IL is INTEGER
    !>
    !>          If RANGE='I', the index of the
    !>          smallest eigenvalue to be returned.
    !>          1 <= IL <= IU <= N, if N > 0.
    !>          Not referenced if RANGE = 'A' or 'V'.
    !>

    IU

    !>          IU is INTEGER
    !>
    !>          If RANGE='I', the index of the
    !>          largest eigenvalue to be returned.
    !>          1 <= IL <= IU <= N, if N > 0.
    !>          Not referenced if RANGE = 'A' or 'V'.
    !>

    M

    !>          M is INTEGER
    !>          The total number of eigenvalues found.  0 <= M <= N.
    !>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
    !>

    W

    !>          W is REAL array, dimension (N)
    !>          The first M elements contain the selected eigenvalues in
    !>          ascending order.
    !>

    Z

    !>          Z is REAL array, dimension (LDZ, max(1,M) )
    !>          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
    !>          contain the orthonormal eigenvectors of the matrix T
    !>          corresponding to the selected eigenvalues, with the i-th
    !>          column of Z holding the eigenvector associated with W(i).
    !>          If JOBZ = 'N', then Z is not referenced.
    !>          Note: the user must ensure that at least max(1,M) columns are
    !>          supplied in the array Z; if RANGE = 'V', the exact value of M
    !>          is not known in advance and can be computed with a workspace
    !>          query by setting NZC = -1, see below.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          JOBZ = 'V', then LDZ >= max(1,N).
    !>

    NZC

    !>          NZC is INTEGER
    !>          The number of eigenvectors to be held in the array Z.
    !>          If RANGE = 'A', then NZC >= max(1,N).
    !>          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
    !>          If RANGE = 'I', then NZC >= IU-IL+1.
    !>          If NZC = -1, then a workspace query is assumed; the
    !>          routine calculates the number of columns of the array Z that
    !>          are needed to hold the eigenvectors.
    !>          This value is returned as the first entry of the Z array, and
    !>          no error message related to NZC is issued by XERBLA.
    !>

    ISUPPZ

    !>          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
    !>          The support of the eigenvectors in Z, i.e., the indices
    !>          indicating the nonzero elements in Z. The i-th computed eigenvector
    !>          is nonzero only in elements ISUPPZ( 2*i-1 ) through
    !>          ISUPPZ( 2*i ). This is relevant in the case when the matrix
    !>          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
    !>

    TRYRAC

    !>          TRYRAC is LOGICAL
    !>          If TRYRAC = .TRUE., indicates that the code should check whether
    !>          the tridiagonal matrix defines its eigenvalues to high relative
    !>          accuracy.  If so, the code uses relative-accuracy preserving
    !>          algorithms that might be (a bit) slower depending on the matrix.
    !>          If the matrix does not define its eigenvalues to high relative
    !>          accuracy, the code can uses possibly faster algorithms.
    !>          If TRYRAC = .FALSE., the code is not required to guarantee
    !>          relatively accurate eigenvalues and can use the fastest possible
    !>          techniques.
    !>          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
    !>          does not define its eigenvalues to high relative accuracy.
    !>

    WORK

    !>          WORK is REAL array, dimension (LWORK)
    !>          On exit, if INFO = 0, WORK(1) returns the optimal
    !>          (and minimal) LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= max(1,18*N)
    !>          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (LIWORK)
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
    !>          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
    !>          if only the eigenvalues are to be computed.
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal size of the IWORK array,
    !>          returns this value as the first entry of the IWORK array, and
    !>          no error message related to LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          On exit, INFO
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>          > 0:  if INFO = 1X, internal error in SLARRE,
    !>                if INFO = 2X, internal error in SLARRV.
    !>                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
    !>                the nonzero error code returned by SLARRE or
    !>                SLARRV, respectively.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
    Jim Demmel, University of California, Berkeley, USA
    Inderjit Dhillon, University of Texas, Austin, USA
    Osni Marques, LBNL/NERSC, USA
    Christof Voemel, University of California, Berkeley, USA
    Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 319 of file sstemr.f.

Author

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